Topological crystalline insulators and superconductors have an excitation gap in the bulk and feature protected gapless or zero-energy modes on their boundaries [1]}, [2]}, [3]}.
These boundary modes are anomalous in the sense that they can only be realized in the presence of a topological bulk.
Crystalline symmetries, such as rotation or inversion symmetry, may protect higher-order topological phases for which anomalous states are located at corners or hinges of the crystal [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
In particular, a \(d\) -dimensional topological crystalline phase of order \(n\)  hosts \((d-n-1)\) -dimensional anomalous states at hinges or corners of the corresponding dimension.
This correspondence between bulk topology and boundary anomaly is a fundamental
aspect of topological insulators and superconductors [28]}, [29]}, [30]}, [31]}, [32]}, [12]}, [34]}, [25]}.
